Elastic Effects in Capillary Flows

About 50 % of the temperature rise occurs near the capillary entrance, z 0.2L. Thus, shortening the capillary length does not decrease the temperature rise due to viscous heating, proportionally. 

Galili and Takserman-Krozer (20) have proposed a simple criterion that signifies when nonisothermal effects must be taken into account. The criterion is based on a perturbation solution of the coupled heat transfer and pressure flow isothermal wall problem of an incompressible Newtonian fluid. 

The pressure drop calculated assuming the relationship Nu = 1.75 (Gz)1 3 for estimating h is smaller than the calculated AP, assuming isothermal flow. For the conditions depicted in Fig. 12.11, at T = 103 s 1 the isothermal pressure drop is about 30 % higher than the measured value. This fact must be taken into account in the design of extrusion dies, so that gross die overdesign can be avoided, as well as in capillary viscometry. 

So far in this chapter we have looked into the viscous phenomena associated with the flow of polymer melts in capillaries. We now turn to the phenomena that are related to melt elasticity, namely (a) swelling of polymer melt extrudates (b) large pressure drops at the capillary entrance, compared to those encountered in the flow of Newtonian fluids and (c) capillary flow instabilities accompanied by extmdate defects, commonly referred to as melt fracture.  

These phenomena have been the subject of intensive study during the last 50 years and still represent major problems in polymer rheology. From a processing point of view they are very important, since melt fracture represents an upper limit to the rate of extrusion, and swelling and the large pressure drops must be accounted for in product considerations and in the design of the die and processing equipment. 

---

Work on designing profile extrusion dies is complicated by the effect known as die swell. In capillary flow, elastic effects cause the diameter of the extrudate to be greater than the capillary diameter. This effect depends on the length of the capillary as well as the processing conditions and must be taken into account when designing extrusion dies. To model such an effect requires a viscoelastic constitutive equation. There is a lack of appropriate models for which data is readily available and this has hindered the use of computer simulation in this field. Nevertheless, a great deal of literature exists on simulation. ... 

Schott, H. (1964) Elastic effects and extrudate distortior s in capillary flow of molten polyethylene resins, J. Polym. Sci, 2, 3791-801. 

As demonstrated, Eq. (7) gives complete information on how the weight fraction influences the blend viscosity by taking into account the critical stress ratio A, the viscosity ratio 8, and a parameter K, which involves the influences of the phenomenological interface slip factor a or ao, the interlayer number m, and the d/Ro ratio. It was also assumed in introducing this function that (1) the TLCP phase is well dispersed, fibrillated, aligned, and just forms one interlayer (2) there is no elastic effect (3) there is no phase inversion of any kind (4) A < 1.0 and (5) a steady-state capillary flow under a constant pressure or a constant wall shear stress. 

Abstract In this contribution, the coupled flow of liquids and gases in capillary thermoelastic porous materials is investigated by using a continuum mechanical model based on the Theory of Porous Media. The movement of the phases is influenced by the capillarity forces, the relative permeability, the temperature and the given boundary conditions. In the examined porous body, the capillary effect is caused by the intermolecular forces of cohesion and adhesion of the constituents involved. The treatment of the capillary problem, based on thermomechanical investigations, yields the result that the capillarity force is a volume interaction force. Moreover, the friction interaction forces caused by the motion of the constituents are included in the mechanical model. The relative permeability depends on the saturation of the porous body which is considered in the mechanical model. In order to describe the thermo-elastic behaviour, the balance equation of energy for the mixture must be taken into account. The aim of this investigation is to provide with a numerical simulation of the behavior of liquid and gas phases in a thermo-elastic porous body. 

Table 7.6 provides a partial reference to studies on the effects of flow on the morphology of polymer blends [Lohfink, 1990 Walling, 1995]. Dispersed phase morphology development has been mainly studied in a capillary flow. To explain the fibrillation processes, not only the viscosity ratio, but also the elasticity effects and the interfacial properties had to be considered. In agreement with the microrheology of Newtonian systems, an upper bound for the viscosity ratio, X, has also been reported for polymer blends — above certain value of X (which could be significantly larger than the... 

The formation of fibrils in poly(carbonate) (PC)/LCP blends have been shown to be enhanced by tbe addition of glass beads. Nano silica acts in the same way. For example, in PC/LCP blends the addition of nano silica results in a reduction of the viscosity. The reduction of viscosity is correlated with the fibrillation of tbe LCP which is promoted by nano silica. In blends of LCP and poly(sulfone) the addition of ca. 5% of nano silica effects the formation of long and perfectly orientated fibrils in the capillary flow. The nano silica forms a network in the matrix that increases the elasticity. This effect is responsible for the improvement of the formation of fibrils. 

In an early paper, Sadowski and Bird (1965) recognised that using a bulk viscosity function for the fluid together with a capillary-hydraulic radius model for the porous media in the manner described above did not take into account any time-dependent elastic phenomena. They suggested that, in a tortuous channel of a porous medium, elastic effects would not be seen provided that the fluid s relaxation time was small compared with the transit time through the contraction/expansion. The fluid would have enough time to readjust to the changing flow conditions. However, if the transit time is small compared with the fluid s relaxation time, then the elasticity of the fluid would have an effect. 

Capillary forces in mixed fluid phase conditions are inversely proportional to the curvature of the interface. Therefore, menisci introduce elasticity to the mixed fluid, and mixtures of two Newtonian fluids exhibit global Maxwellian response. For more details see Alvarellos [1], his behavior is experimentally demonstrated with a capillary tube partially filled with a water droplet. The tube is tilted at an angle (3 smaller than the critical angle that causes unstable displacement. Then, a harmonic excitation is applied to the tube in the axial direction. For each frequency, the amplitude of the vibration is increased until the water droplet becomes unstable and flows in the capillary. Data in Figure 3 show a minimum required tube velocity between 40 and 50 Hz. This behavior indicates resonance of the visco-elastic system. The ratio of the relaxation time and characteristic time for pure viscous effect is larger than 11.64. 

Indeed, as fluid flows, foam channels closed above grow in thickness at the bottom, thus creating an increasing counteraction to the gravitational force, which slows down the outflow until equilibrium is attained [214]. It should be noted that this effect is possible only in closed deformable channels with negative curvature, which are typical of foam. According to [324], the capillary rarefaction is a characteristic of the foam compressibility and determines its elastic resistance to the strain caused by the liquid redistribution. 

In normal capillary rheometry for polymer melts, the flowing stream exits into the atmosphere, and the driving static pressure in the reservoir is taken to he AP. In such cases, end effects involving viscous and elastic deformations at the entrance and exit of the capillary should be taken into account when calculating the true shear stress at the capillary wall, particularly if the ratio of capillary length to radius L/R) is small. 

---